Without the use of cheats, no they can't. If a toddlers hygiene, hunger and social needs fall too low a van will arrive outside the sim's house. After lecturing the parents of the toddler the Social Worker will then pick the toddler up, put them in the van and drive away. As well as the neglected toddler all other sims younger than a teenager will get taken away too.
However if you type in 'boolProp testingcheatsenabled true' without apostrophes into the cheat bar (opened by pressing the ctrl, shift and c keys at the same time) and then shift-clicking the sim in question, a series of possible commands pops up. Browse through the options until you find 'spawn'. Click that and then select 'Rodney's Death Creator'. A tombstone will appear next to your Sim (can be moved in buy mode) when you click (not shift-click) the tombstone you are given a series of options of all possible ways to kill a Sim. This enabled any sim of any age to die any death. That means a toddler can die of Old Age or an child of fire. This is great for making movies.
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No
Yes you can die on Sims 2 PSP
You can only tickle toddlers and babies. Click on them, and select Tickle.
Yes they grow up from toddlers to children.
It died of disease